### Nejc Širovnik

Department of Mathematics and Computer Science, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, 2000 Maribor, Slovenia
e-mail: nejc.sirovnik@uni-mb.si

Abstract.   In this paper functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove, for example, the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X into itself and let A(X) ⊂ L(X) be a standard operator algebra. Suppose there exist linear mappings D,G:A(X) → L(X) satisfying the relations D(A3)=D(A2)A+A2G(A),G(A3)=G(A2)A+A2D(A) for all A A(X). In this case there exists B L(X) such that D(A)=G(A)=[A,B] holds for all A A(X).

2010 Mathematics Subject Classification.   16N60, 46B99, 39B42.

Key words and phrases.   Prime ring, semiprime ring, Banach space, standard operator algebra, derivation, Jordan derivation, Jordan triple derivation.

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DOI: 10.3336/gm.47.1.07

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